Exercise 10, § 10

Let $k$ be a field, $A$ a regular local $k$-algebra of dimension $d$, such that the extension $k \to \kappa_A$ is trivial. The $A$-modules $H^d_A(A)$ and $A' = \mathrm{Hom}_k^{cont}(A, k)$ are Matlis modules, hence isomorphic; we propose to make explicit an isomorphism between these two modules.

(a) Let $x = (x_1, \ldots, x_d)$ be a system of coordinates of $A$. For every integer $p \geqslant 0$, denote by $A_p$ the $k$-algebra $A/(x_1^p, \ldots, x_d^p)$. The classes of the monomials $x_1^{a_1} \ldots x_d^{a_d}$ with $0 \leqslant a_i < p$ form a basis of $A_p$ over $k$; denote by $\rho_p$ the linear form on $A_p$ which has value 1 on the class of $x_1^{p-1} \ldots x_d^{p-1}$ and 0 on the other elements of the basis. Prove that the $k$-linear mapping $\tilde{\rho}_p : A_p \to \mathrm{Hom}_k(A_p, k)$ deduced from the bilinear form $(a, b) \mapsto \rho_p(ab)$ is an isomorphism (one may reason directly, or use exerc. 10 of § 5).

(b) For $q \geqslant p$, let $u_{qp} : A_p \to A_q$ be the $A$-linear homomorphism induced by the homothety of ratio $(x_1 \ldots x_d)^{q-p}$, and let $\pi_{pq} : A_q \to A_p$ be the mapping of passing to the quotient; prove that one has $\tilde{\rho}_q \circ u_{qp} = {}^t \pi_{pq} \circ \tilde{\rho}_p$. Deduce by passing to the inductive limit, taking into account exerc. 8, c), an $A$-linear isomorphism $\tilde{\rho} : H^d_A(A) \to A'$.

(c) Suppose $d = 1$, so that $A$ is a discrete valuation ring, and $x_1 = x$ a uniformizer of $A$. Show that the choice of the uniformizer $x$ permits identifying $H^1_A(A)$ with $K/A$; the isomorphism $\rho : K/A \to A'$ is then defined by $\rho(\varphi)(a) = \mathrm{Res}(a \varphi)$, where $\mathrm{Res} : K/A \to k$ is the projection on the factor $kx^{-1}$ in the canonical decomposition $K/A = \bigoplus_{n \geqslant 1} kx^{-n}$.